Discrete Morse Theory and a Reformulation of the K(\pi,1)-conjecture

TL;DR

This paper uses discrete Morse theory and homotopy theory to provide an alternative proof of a key conjecture in Artin groups and introduces new models for classifying spaces and chain complexes for monoid homology.

Contribution

It offers a novel proof of the K(,1)-conjecture for Artin groups and presents new models for classifying spaces and chain complexes for monoid homology.

Findings

Alternative proof of the K(,1)-conjecture using discrete Morse theory.

A new model for the classifying space of an Artin monoid.

A small chain complex for computing monoid homology.

Abstract

A recent theorem of Dobrinskaya states that the K(\pi,1)-conjecture holds for an Artin group G if and only if the canonical map from BM to BG is a homotopy equivalence, where M denotes the Artin monoid associated to G. The aim of this paper is to give an alternative proof by means of discrete Morse theory and abstract homotopy theory. Moreover, we exhibit a new model for the classifying space of an Artin monoid, in the spirit of Charney, Meier and Whittlesey, and a small chain complex for computing its monoid homology, similar to the one of Squier.